He told the BBC's Panorama his party had to listen to Remain voters about their concerns.
The turmoil started when May chose to hold snap elections to gain more seats for her party, which already had majority in parliament, to strengthen her position in the upcoming Brexit talks scheduled for June 19.
And significant changes to her policy platform are likely if she survives.
Prime Minister Theresa May reappointed most of her ministers on Sunday but brought a Brexit campaigner and party rival into government to try to unite her Conservatives after a disastrous election sapped her authority, days before Brexit talks begin.
May's Conservatives unexpectedly lost their majority in parliament in snap vote, causing political chaos ahead of Brexit talks with EU.
The Conservatives won 318 seats in last week's election, eight short of a majority, and therefore need the support of at least one other party to pass key legislation in Parliament.
Ms May faced Conservative MPs at a meeting of the party's 1922 Committee.
Earlier, Foster said the talks were going well: "We hope soon to be able to bring this work to a successful conclusion."May made clear the Brexit negotiations would begin on schedule despite the political uncertainty at home."I confirmed to President Macron that the timetable for the Brexit negotiation remains on course and will begin next week", May said after her meeting with the new French leader, who will be a key player in the Brexit talks."I think there is a unity of goal among people in the United Kingdom".
Addressing a meeting of backbenchers, the PM reportedly said she would serve as "long as you want me to do".
The Daily Telegraph reported cabinet ministers have opened back-channel talks to senior Labour lawmakers to secure a cross-party agreement on Brexit.
Mr Corbyn warned against a "coalition of chaos" between the Conservatives and the DUP and said Labour was ready to provide "strong and stable leadership", mocking two slogans used by Mrs May during the campaign.
With the two-year clock on Brexit ticking away since March, when a letter from May formally started proceedings, Barnier dismissed suggestion of postponing the negotiations and said such a delay would only prompt further instability.
The UK is keen to negotiate a trade deal at the same time as sorting out an exit payment and agreeing the future of European Union citizens living in Britain.
Following the disappointing results for the Conservative Party and the promising results for Labour in the June 8 general elections, many Britons are calling on May to resign and give way to the Labour Party to form a new government.
Opponents of a sharp break include Ruth Davidson, the leader of the Scottish Conservatives who helped the party win 12 more seats in Scotland in contrast to losses elsewhere.
The pound slid to its lowest level for almost two months after the vote, but the fall was much less severe than the one sparked by the Brexit vote in June 2016.
May's spokesman said it remained government policy to cut net migration to under 100,000 and Brexit Minister David Davis also said walking away without securing a deal with the remaining 27 European Union states remained a possibility.
Amid calls from some MPs for the Conservatives to rethink their Brexit strategy, he said there was a "clear consensus" for leaving the single market and ending free movement while retaining the "maximum access" to European Union markets and maintaining co-operation in key areas such as science.
In an article in the Belfast Telegraph, Foster listed three priorities, including getting Northern Ireland's devolved power-sharing government at Stormont working again.
"Our position remains that we want to get the best possible deal for the United Kingdom", he said.
"We will use the position we find ourselves in to do as we promised".
However, George Osborne, a former finance minister who is now editor of London's Evening Standard newspaper and a vocal critic of May, said she appeared a "dead woman walking".
Britain's descent into political crisis just days before the Brexit talks begin has sapped confidence amongst business leaders and infuriated bosses who were already grappling with the fallout from the vote to leave the EU.